Theorem 2omotap 7406

Description: If there is at most one tight apartness on 2_o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)

Assertion

Ref	Expression
2omotap	⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)

Proof of Theorem 2omotap

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2omotaplemst 7405	. . . . 5 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝑥 = {∅}) → 𝑥 = {∅})
2	1	ex 115	. . . 4 ⊢ (∃*𝑟 𝑟 TAp 2o → (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
3		df-stab 833	. . . 4 ⊢ (STAB 𝑥 = {∅} ↔ (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
4	2, 3	sylibr 134	. . 3 ⊢ (∃*𝑟 𝑟 TAp 2o → STAB 𝑥 = {∅})
5	4	adantr 276	. 2 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
6	5	exmid1stab 4268	1 ⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 832   = wceq 1373  ∃*wmo 2056   ⊆ wss 3174  ∅c0 3468  {csn 3643  EXMIDwem 4254  2_oc2o 6519   TAp wtap 7396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-tr 4159  df-exmid 4255  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-1o 6525  df-2o 6526  df-pap 7395  df-tap 7397

This theorem is referenced by:  exmidmotap  7408